Resumen de Calculation of Force Constants for Structures with Tetrahedral Symmetry
This paper develops an analytical method for calculating the transferable force constants in symmetrically complete coordinates directly from the nontransferable force constants in independent coordinates allowing a direct treatment of the redundancy problem. In calculating the force constants for structures with tetrahedral symmetry, the interdependence of the tetrahedral angles must be considered to be a constraint on the potential energy function. The methane molecule is chosen as the simplest example of a tetrahedron. The potential energy of methane is a function of six equally weighted tetrahedral angles constrained such that the sum of the changes in angle over all six angles is zero. The calculations with independent coordinates are done with two different parameterizations, first with five tetrahedral angles, and second with three tetrahedral angles and two dihedral angles. These are the two most useful and common parameterizations used in molecular orbital calculations. The methane tetrahedron is modeled with seven force constants; this paper demonstrates that only five independent variables Kr, Krr, (KrA-KrV), (Kθ-KV) and (Kθ-KA) are needed. This agrees with the fact that the entire vibrational spectrum of methane can be calculated from only five force constants. Sample Gaussian 90 calculations are done with Slater type orbital 3-21G* basis sets and compared with neutron dispersion data for diamond, silicon, germanium, and tin. This method can be extended to other constrained systems such as rings.
